# Indefinite Integration of Rational Functions

How to solve the indefinite integration of rational functions problems: examples and their solutions.

## Example 1

To integrate a fraction,

change the fraction to make (constant)/(linear) form.

If (numerator's order) ≥ (denominator's order),

then use long division or synthetic division

to make (constant)/(linear) form.

Dividing polynomials (Long division)

Synthetic division

So ∫ (*x*^{2} - 3*x* + 5)/(*x* - 1) *dx* = ∫ ( *x* - 2 + 3/[*x* - 1] ) *dx*.

Integrate the given integral.

Then (given) = (1/2)*x*^{2} - 2*x* + 3 ln |*x* - 1| + *C*.

Linear change of variable rule

Indefinite integration of 1/*x*

## Example 2

See [3*x* - 2] / [*x*(*x* - 1)].

(numerator's order) < (denominator's order)

So you don't need to use

long division or synthetic division.

But the denominator is not in linear form.

So use partial fraction decomposition

to make (constant)/(linear) form.

Set [3*x* - 2] / [*x*(*x* - 1)] = *A*/*x* + *B*/(*x* - 1).

Then compare the numerators on both sides

to find *A* and *B*.

*A* = 2, *B* = 1.

So [3*x* - 2] / [*x*(*x* - 1)] = 2/*x* + 1/(*x* - 1).

So (given) = ∫ [2/*x* + 1/(*x* - 1)] *dx*

Integrate the given integral.

Then (given) = 2 ln |*x*| + ln |*x* - 1| + *C*.

Linear change of variable rule

Indefinite integration of 1/*x*